Convolution by L¹ on Lᵖ of a locally compact group #
Let μ be a left-invariant measure on a topological group G which is locally finite and inner
regular for compact sets, for instance a regular left Haar measure on a locally compact group. An
integrable kernel h with values in a normed field 𝕜, such as ℝ or ℂ, acts on the
𝕜-normed space Lᵖ(G, μ; E), for 1 ≤ p < ∞, by left convolution
h ⋆ f = ∫ y, h(y) • L_y f dμ(y), (L_y f)(x) = f (y⁻¹ * x),
an Lᵖ-valued Bochner integral of left translates of f. This is the operator form of the
classical convolution (h ⋆ f)(x) = ∫ y, h(y) • f(y⁻¹ * x) dμ(y); the two agree almost
everywhere for compactly supported continuous h and f, by Fubini's theorem for compactly
supported continuous integrands, which needs no σ-finiteness. The group need not be abelian,
and neither σ-finiteness of μ nor second countability of G is assumed. The integrand is
almost everywhere strongly measurable because y ↦ L_y f is continuous
(MeasureTheory.Lp.continuous_leftTranslation_apply), and an integrable h lives on a
σ-finite set (MeasureTheory.AEFinStronglyMeasurable.aestronglyMeasurable_smul).
Since the definition never evaluates h ⋆ f at a point, its algebraic properties need no Fubini
theorem. Left translation passes through the integral, and a left translation of the variable
y turns L_g (h ⋆ f) into (L_g h) ⋆ f, while on a locally compact group with a regular Haar
measure right translation passes directly to the second factor, R_g (h ⋆ f) = h ⋆ (R_g f).
Applied to the operator u ↦ u ⋆ f, which is continuous and linear in u, left equivariance
gives associativity. The positive cone of real Lᵖ is closed, so convolution of nonnegative
functions is nonnegative. Integration over G is a continuous linear functional on L¹ invariant
under left translation, so integrals multiply. In particular, convolution preserves nonnegative
L¹ functions of integral one.
On a locally compact group with a regular Haar measure, the modular inversion
(J f)(x) = c(x) • f x⁻¹ of L¹, where c is the modular character, reverses convolution:
J (h ⋆ k) = J k ⋆ J h. The identity is checked against set integrals for compactly supported
continuous h and k, and extended to all of L¹ by density and continuity
(MeasureTheory.Lp.dense_setOf_continuous_hasCompactSupport).
The integrated form ContRepresentation.integratedOperatorL1 is the corresponding construction
for an arbitrary strongly continuous representation in place of left translation; it is stated
for an additive, not necessarily commutative, monoid and reaches multiplicative groups only
through Additive G, whereas this file works with the multiplicative group G directly. Similarly,
TauCeti.convolutionLp is the additive version of this construction on a finite-dimensional real
vector space with an additive Haar measure.
Main definitions #
MeasureTheory.Lp.convolutionL μ p hp: the𝕜-bilinear map(h, f) ↦ h ⋆ ffromL¹(G, μ; 𝕜)andLᵖ(G, μ; E)toLᵖ(G, μ; E), of norm at most one.
Main results #
MeasureTheory.Lp.convolutionL_apply: the definingLᵖ-valued Bochner integral.MeasureTheory.Lp.integrable_smul_leftTranslation: its integrand is integrable.MeasureTheory.Lp.norm_convolutionL_apply_le: Young's inequality‖h ⋆ f‖_p ≤ ‖h‖₁ ‖f‖_p.MeasureTheory.Lp.leftTranslation_convolutionL:L_g (h ⋆ f) = (L_g h) ⋆ f.MeasureTheory.Lp.rightTranslation_convolutionL:R_g (h ⋆ f) = h ⋆ (R_g f).MeasureTheory.Lp.convolutionL_assoc: associativity,(h ⋆ k) ⋆ f = h ⋆ (k ⋆ f).MeasureTheory.Lp.convolutionL_nonneg:h ⋆ f ≥ 0whenh ≥ 0andf ≥ 0.MeasureTheory.Lp.integral_convolutionL:∫ (h ⋆ f) = (∫ h) • ∫ f.MeasureTheory.Lp.convolutionL_toL1_ae_eq: for compactly supported continuoushandf, the convolutionh ⋆ fis represented byx ↦ ∫ y, h y • f (y⁻¹ * x) dμ.MeasureTheory.Lp.modularInversion_convolutionL: the modular inversion reverses convolution,J (h ⋆ k) = J k ⋆ J h.
References #
- G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press (2016), §2.5 and §3.2.
- B. Bekka, P. de la Harpe, A. Valette, Kazhdan's Property (T), proof of Theorem G.3.1.
The integrand y ↦ h(y) • L_y f of a convolution is integrable. Together with
convolutionL_apply, this lets continuous linear maps pass through the integral defining
h ⋆ f.
Convolution on a group G by an integrable kernel, as the bilinear map
L¹(G, μ; 𝕜) →L Lᵖ(G, μ; E) →L Lᵖ(G, μ; E) sending (h, f) to the Lᵖ-valued Bochner
integral h ⋆ f = ∫ y, h(y) • L_y f dμ(y) of the left translates (L_y f)(x) = f (y⁻¹ * x).
Instances For
The convolution h ⋆ f is the Lᵖ-valued integral of the left translates of f weighted
by h.
The convolution of the L¹ class of an integrable function h with f.
Young's inequality for an L¹ kernel: ‖h ⋆ f‖_p ≤ ‖h‖₁ ‖f‖_p.
Convolution has operator norm at most one as a bilinear map.
Convolution commutes with left translation: L_g (h ⋆ f) = (L_g h) ⋆ f.
Associativity of convolution: (h ⋆ k) ⋆ f = h ⋆ (k ⋆ f) for L¹ kernels h and
k.
The convolution of nonnegative functions is nonnegative.
The integral of a convolution is the product of the integrals: ∫ (h ⋆ f) = (∫ h) • ∫ f.
For compactly supported continuous h and f, the convolution h ⋆ f is represented almost
everywhere by the pointwise convolution x ↦ ∫ y, h y • f (y⁻¹ * x) dμ.
The modular inversion reverses convolution: J (h ⋆ k) = J k ⋆ J h for h, k ∈ L¹,
where (J f)(x) = c(x) • f x⁻¹ and c is the modular character.
Convolution commutes with right translation on the second factor:
R_g (h ⋆ f) = h ⋆ (R_g f).